Calculus (Differentiation and Integration)
What the domain tests
Calculus tests differentiation and integration of powers, trigonometric and exponential expressions, and applications to gradients, stationary points, rates and areas. MSCE Additional Mathematics expects correct notation and careful evaluation of limits in definite integrals.
Core rules
- If y=x^n, dy/dx=nx^{n-1}. Differentiate sums term by term. Chain rule: d/dx[f(g(x))]=f'(g(x))g'(x). Product and quotient rules apply to products and quotients.
- d/dx(sinx)=cosx; d/dx(cosx)=-sinx; d/dx(e^x)=e^x; d/dx(lnx)=1/x.
- At a stationary point dy/dx=0. Use the second derivative or a sign table to classify maximum and minimum.
- ∫x^n dx=x^{n+1}/(n+1)+C for n≠-1; ∫(1/x)dx=ln|x|+C. A definite integral gives signed area; geometric area between a curve and the x-axis requires splitting where the curve crosses the axis.
Worked example
Find the area under y=x^2-4x+3 between x=1 and x=3. ∫_1^3(x^2-4x+3)dx=[x^3/3-2x^2+3x]_1^3. At x=3: 9-18+9=0. At x=1: 1/3-2+3=4/3. The definite integral is 0-4/3=-4/3. The curve lies below the axis on this interval, so the geometric area is 4/3 square units.
Traps
Omitting +C in indefinite integrals; treating signed area as geometric area; differentiating before simplifying; checking only stationary points and forgetting endpoints.